ATM Vol & Correlation

• ATM Vol can be represented through the cosine rule. The longer the length the higher the ATM vol.

  $\sigma_{EURGBP}^2 =$$\sigma_{EURUSD}^2 +$$\sigma_{GBPUSD}^2 - 2(\sigma_{EURUSD} \cdot \sigma_{GBPUSD} \cdot \rho)$

  The angle (theta) is the inverse cosine of the correlation between EURUSD and GBPUSD

  
• As the correlation between pairs get higher, the angle between the implied vol triangle gets smaller. At 0% correlation the angle is 90

  
• If the angle is 0, the lines will be on top of each other. If EURUSD and GBPUSD moves perfectly together, EURGBP will just be static
• If EURUSD ATM vol is 10%, GBPUSD ATM vol is 10%, the relationship between the correlation and cross-volatility (EURGBP vol) is:

  
• Correlation and cross-volatility move in opposite directions

  • Higher correlation between major pairs → tighter angle in ATM volatility triangle → lower cross volatility
  • Lower correlation between major pairs → wider angle in ATM volatility triangle → higher cross volatility
• Sometimes traders transact delta hedges and split the risk into two separate pairs. I.e if a trader long EUR/JPY delta and short CHF/JPY delta, they may sell EUR/CHF spot and split it into short EUR/JPY and long CHF/JPY spot trades, with offsetting JPY amounts
• Volatility cone is a common way to visualize how cheap/expensive ATM curve compared to historical curve

  
• If the slope of ATM curve is steep, it might be attractive to do calendar spreads. The weighting can be:

  • Vega-neutral (to reduce P&L volatility from parallel moves in the ATM curve)
  • Gamma-neutral (to reduce the P&L volatility caused by spot moves)
• At longer tenors, ATM implied vol is relatively more impacted by interest rates vol than spot vol
• In some pairs, ATM vol around 1yr tenor can exhibit seasonality due to corporate hedging. Corporate hedge structures mostly are net sell vega causing vol to be suppressed